[Paper Review] A Projective Surface of Degree Eight with 168 Nodes

Stephan Endraß|ArXiv.org|Jul 19, 1995

Algebraic Geometry and Number TheoryMathematics3 references22 citations

TL;DR

This paper constructs a degree-eight algebraic surface in complex projective 3-space with exactly 168 ordinary double points (nodes), demonstrating that the maximum number of nodes for an octic surface, μ(8), is at least 168. Using a seven-parameter family of D₈-symmetric octic surfaces and applying the Segre-trick via dihedral symmetry and computer algebra, the authors explicitly construct a rigid surface invariant under D₈ × ℤ₂, confirming 168 nodes and closing the gap between previous lower bounds and Miyaoka's upper bound of 174.

ABSTRACT

The estimate for the maximal number of ordinary double points of a projective surface of degree eight is improved to $168\leqμ(8)\leq 174$ by constructing a projective surface of degree eight with 168 nodes.

Motivation & Objective

• To improve the known lower bound for the maximum number of nodes μ(8) on a degree-eight surface in ℙ³.
• To construct an explicit example of an octic surface with 168 nodes, surpassing prior bounds of 160.
• To demonstrate that such a surface exists within a D₈-symmetric family and is rigid, using computational algebraic geometry.
• To confirm that no intermediate numbers of nodes (161–167) exist in the family, ruling out their existence via the construction.

Proposed method

• Define a D₈-invariant polynomial P as a product over eight planes, vanishing to order two along their 28 intersection lines.
• Construct a second D₈-invariant polynomial Q as a square of a quartic form in x²+y², z, w, with seven free parameters.
• Form the surface F = P - Q, which generically has 112 nodes from the second-order vanishing of P and Q on the 28 lines.
• Impose symmetry constraints (c=f=h=0, e=−1) to enhance symmetry to D₈ × ℤ₂ and enable the Segre-trick via plane quartic quotients.
• Use computer algebra (Maple V R3 and MACAULAY) to compute the Hessian determinants at candidate singular points to verify 168 nodes.
• Apply a lemma to rule out additional singularities outside the eight symmetry planes, proving the surface has exactly 168 isolated nodes.

Experimental results

Research questions

• RQ1 Can a degree-eight surface in ℙ³ have more than 160 nodes, and what is the exact maximum possible number?
• RQ2 Does a D₈-symmetric family of octic surfaces contain a surface with 168 nodes, and is it rigid?
• RQ3 Can the Segre-trick be applied to construct such a surface via a quotient by a reflection group?
• RQ4 Are there any surfaces with 161 to 167 nodes in the same family, and if so, are they realizable?
• RQ5 Can the absence of singularities outside the symmetry planes be rigorously proven using Hessian determinants and symmetry arguments?

Key findings

• The constructed octic surface X₈ has exactly 168 nodes, confirming that μ(8) ≥ 168.
• The surface is rigid, as computed by D. van Straten using MACAULAY, meaning it has no moduli.
• The surface is invariant under the group D₈ × ℤ₂, and thus is an eightfold cover of a quartic surface with 13 nodes.
• The 168 nodes arise from 56 additional nodes computed via Hessian determinants at specific points, beyond the initial 112.
• No singularities exist outside the eight symmetry planes E_j or the line L, as proven by contradiction using a lemma on orbits of nodes.
• The family of surfaces contains no surfaces with 161 to 167 nodes, so the gap between 160 and 168 remains unbridgeable in this construction.